A numerical comparison of some heuristic stopping rules for nonlinear Landweber iteration

TL;DR

This paper numerically compares heuristic stopping rules for nonlinear Landweber iteration, assessing their effectiveness in various inverse problems where noise level estimates are unavailable.

Contribution

It provides the first comprehensive numerical analysis of heuristic stopping rules for nonlinear iterative regularization methods across diverse practical inverse problems.

Findings

Certain heuristic rules perform reliably across different nonlinear problems.

Performance varies depending on the problem type and noise conditions.

Guidelines for choosing stopping rules in nonlinear inverse problems are suggested.

Abstract

The choice of a suitable regularization parameter is an important part of most regularization methods for inverse problems. In the absence of reliable estimates of the noise level, heuristic parameter choice rules can be used to accomplish this task. While they are already fairly well-understood and tested in the case of linear problems, not much is known about their behaviour for nonlinear problems and even less in the respective case of iterative regularization. Hence, in this paper, we numerically study the performance of some of these rules when used to determine a stopping index for Landweber iteration for various nonlinear inverse problems. These are chosen from different practically relevant fields such as integral equations, parameter estimation, and tomography.